On the Standardization Theorem for λβη-calculus

Transcription

1 On the Standardization Theorem for λβη-calculus Ryo Kashima Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Meguro, Tokyo , Japan. September 2001 Abstract We present a new proof of the standardization theorem for λβη-calculus, which is performed by inductions based on an inductive definition of βηreducibility with a standard sequence. 1 Introduction The standardization theorem is a fundamental theorem in reduction theory of λ- calculus, which states that if a λ-term M β-(or βη-)reduces to a λ-term N, then there is a standard β-(or βη-)reduction sequence from M to N. In [3], the author gave a simple proof of the theorem for β-reduction. This paper extends the result to βη-reduction: we give a simple proof of the (weak) standardization theorem for βη-reduction. There have been some proofs of the standardization theorem in literature (e.g., [1, 2, 4, 5, 6, 7]). Compared with these, a feature of the presented proof is that we use an inductive definition (formal theory) of βη-reducibility with a standard sequence. In virtue of this definition, all the proof can be performed by easy inductions. Applications of this method to proofs of other theorems (e.g., the leftmost reduction theorem for βη-reduction) and other calculi (e.g., term rewriting systems) are future studies. 2 Preliminaries λ-terms are constructed by application and abstraction from variables (λ-terms of the form (MN) and (λx.m) are called an application and an abstraction respectively). Capital letters A, B,... denote arbitrary λ-terms, and small letters x, y,... denote arbitrary variables. M[x := N] denotes the result of substituting N for all the free occurrences of x in M with adequate renaming of bound variables. The set 1

2 of free variables in M is written by FV(M). We identify λ-terms that are mutually obtained by renaming of bound variables. In successive abstractions or applications, parentheses are omitted as follows: for example, λxyz.a = λx.(λy.(λz.a)), and ABCD = ((AB)C)D. (See, e.g., [1] for the basic notion and terminology.) The binary relation βη (one-step βη-reducibility) on the set of λ-terms are defined as usual: ( ((λx.a)b) ) βη ( (A[x := B]) ) and ( (λx.(cx)) ) βη ( C ) where x FV(C). The subterms (λx.a)b and λx.(cx) are called βη-redexes where the former and the latter are also called a β-redex and an η-redex respectively. Let R be an occurrence of a βη-redex in a λ-term M. We write M R βη N if and only if N is obtained from M by contracting the redex R. The relation l βη (one-step leftmost βη-reducibility) is defined as usual: M l βη N if and only if M R βη N for the leftmost βη-redex R in M, where a redex occurrence R 1 in M is said to be to the left of another redex occurence R 2 in M if they occur as follows. M = ( R 1 R 2 ) or M = ( ( R 2 ) ) }{{} R 1 The binary relations βη and l βη are the reflexive transitive closures of βη and l βη respectively. Suppose that M R βη N and that P and Q are subterm occurrences of M and N respectively. We write P Q if Q is a one step residual of P. For example, if ( ( {}} ) { ) M = P 1 (λx. ( P 3 )) ( P }{{} 4 ) P }{{} 5 A B P 2 Q 2 ( ( {}} ) { ) N = P 1 ( P 3 )[x := ( P }{{} 4 )] P }{{} 5 A B where R = (λx.a)b P 2 and P 3 x, then P 1 P 1, P 2 Q 2, P 3 P 3 [x := B], P 4 each copy of P 4 in A[x := B], and P 5 P 5. (Note that the redex (λx.a)b has no residual.) Suppose that ( X ) βη βη ( Y ). We say that Y is a residual of X if X Z 1 Z n Y for some Z 1,..., Z n (n 0). A βη-reduction sequence R M 1 R 1 βη M 2 2 βη Rn 1 βη M n is called weakly standard if the following condition is satisfied. 2

3 i [ R i+1 is not a residual of a βη-redex occurrence P in M i such that P is to the left of R i ] Moreover, the sequence is called strongly standard if the following condition is satisfied. i, j > i [ R j is not a residual of a βη-redex occurrence P in M i such that P is to the left of R i ] (The distinction between weak and strong appeared in [2]. Note that these two conditions are equivalent if η-reduction does not exist.) For example, the sequence λx.(iy(ix)) Iy βη λx.(y(ix)) Ix βη λx.(yx) λx.(yx) βη y is weakly and strongly standard; the sequence λx.(iy(ix)) Ix βη λx.(iyx) λx.(iyx) βη Iy Iy βη y is not strongly standard but weakly standard; and the sequence λx.(iyx) Iy βη λx.(yx) λx.(yx) βη y is neither strongly nor weakly standard, where I = λz.z. In this paper, we present a simple proof of the theorem: Theorem 2.1 (Weak Standardization Theorem) If M βη N, then there is a weakly standard βη-reduction sequence from M to N. Other important theorems, which have been proved in the literature (e.g., [4, 6]), are there: (Strong Standardization Theorem) If M βη N, then there is a strongly standard βη-reduction sequence from M to N. (Leftmost Reduction Theorem) If M βη N and N is a βη-normal form, then M l βη N. We will discuss them in Section 4. 3 Proof We define a formal theory which proves formulas of the forms A hap B and A st B. Axioms (Id) A hap A. (β) (λx.a)bc 1 C n hap A[x:=B]C 1 C n, where n 0. 3

4 Rules A hap B B hap C A hap C (Tr) L hap x L st x (Var) L hap AB A st C B st D L st CD L hap λx.a A st B L st λx.b L hap λx.a A st Bx with the proviso: L st B x FV(B) and B is not an abstraction. We write A hap B (or A st B) if and only if the formula A hap B (or A st B, respectively) is provable in this system. ( hap and st stand for head reduction in application and standard.) Theorem 3.1 If M hap N, then M l βη N. If M st N, then there is a weakly standard βη-reduction sequence from M to N. Proof By induction on the proof of M hap N or M st N. Here we show the only nontrivial case: The proof is.. M hap λx.a A st Nx M st N where x FV(N) and N is not an abstraction. By the induction hypotheses, there are a leftmost reduction sequence L from M to λx.a, and a weakly standard reduction sequence S from A to Nx. We will show that there is a weakly standard reduction sequence S + from λx.a to N; then the concatenation of L and S + is the required standard sequence from M to N. We will explain, based on an example, the definition of the sequence S +. If a λ-term P is of the form P x and x FV(P ), then we say that P is an η-body for x. Suppose that the weakly standard reduction sequence S is where A R 1 βη A R 2 βη Bx R 3 βη (λy.(cy))x (λy.(cy))x βη Cx R 5 βη Nx Bx, (λy.(cy))x, Cx, and Nx are η-bodies for x; 4

5 A is not an η-body for x; B R 3 βη λy.(cy); and C R 5 βη N. Note that R 3 = B; otherwise, reduction of R 3 cannot make λy.(cy) if B is an application, and S is not weakly standard if B is an abstraction. Then, the weakly standard sequence S + is defined as follows. λx.a R 1 βη λx.a R 2 βη λx.(bx) λx.(bx) βη B R 3 βη λy.(cy) λy.(cy) βη C R 5 βη N. Lemma 3.2 (1) If M hap N, then MP hap NP. (2) If L hap M st N, then L st N. (3) If M hap N, then M[z := P ] hap N[z := P ]. Proof (1) By induction on the proof of M hap N. (2) If M st N is proved, then there must be a premise M hap P for certain P. From this and the assumption L hap M, we can infer L hap P by the rule (Tr). Then L st N is proved by the rule that infers M st N. (3) By induction on the proof of M hap N. Lemma 3.3 If M st N and P st y, then M[z := P ] st N[z := y]. Proof By induction on the proof of M st N. We divide cases according to the last inference of the proof. The substitution [z := P ] and [z := y] will be represented by [P ] and [y] for short. (Case 1): The last inference is where N = x z. In this case, we have (Case 2): The last inference is M hap x M st x (Var) and Lemma 3.2(3) M[P ] hap x (Var) M[P ] st x. M hap z M st z 5 (Var)

6 where N = z. In this case, we have and Lemma 3.2(3) M[P ] hap P M[P ] st y.. assumption of the lemma P st y Lemma 3.2(2) (Case 3): The last inference is. σ M hap AB A st C M st CD B st D where N = CD. In this case, we have and Lem.3.2(3). i.h. for σ M[P ] hap (AB)[P ] A[P ] st C[y] M[P ] st (CD)[y].. i.h. for τ B[P ] st D[y] (Case 4): The last inference is where N = λx.b. If x z, then we have. σ M hap λx.a A st B M st λx.b and Lem.3.2(3). i.h. for σ M[P ] hap (λx.a)[p ] A[P ] st B[y] M[P ] st (λx.b)[y] where we assume that x FV(P, y) (otherwise we rename x). If x = z, then (λx.x)[z := Z] = λx.x, and the induction hypothesis is not necessary. (Case 5): The last inference is. σ M hap λx.a A st Nx M st N where x FV(N) and N is not an abstraction. If x z, then we have and Lem.3.2(3). i.h. for σ M[P ] hap (λx.a)[p ] A[P ] st (Nx)[y] M[P ] st N[y] where we assume that x FV(P, y) (otherwise we rename x). Note that (Nx)[y] = (N[y])x and the proviso for N[y] is satisfied. If x = z, then the induction hypothesis is not necessary. 6

7 Lemma 3.4 If L st (λx 1 x m.m)y 1 y m F 1 F n, then L st M[x 1 :=y 1,..., x m := y m ]F 1 F n, where m 1, n 0 and x 1,..., x m are distinct. (Note that [x 1 := y 1,..., x m := y m ] is a simultaneous substitution, and this will be represented by [ y] for short.) Proof By induction on the proof of L st (λx 1 x m.m)y 1 y m F 1 F n. We divide cases according to the last inference of the proof. (Case 1): n = 0, and the proof is L hap AB. σ A st (λx 1 x m.m)y 1 y m 1 L st (λx 1 x m.m)y 1 y m. B st y m If m 2, then by the induction hypothesis for σ, there is a proof. σ A st (λx m.m)[ y ] where [ y ] = [x 1 := y 1,..., x m 1 := y m 1 ]. If m = 1, then we define σ = σ. In any case, the proof σ must be of the form. φ A hap λx m.c. ψ C st M[ y ] A st λx m.(m[ y ]). (Note that x m {y 1,..., y m 1 }; otherwise we rename x m.) Then we have where ( ) is. ( ). ( ) L hap C[x m :=B] C[x m :=B] st M[ y] Lem. 3.2(2) L st M[ y] L hap AB. φ A hap λx m.c AB hap (λx m.c)b Lem.3.2(1) (axiom β) (λx m.c)b hap C[x m :=B] L hap C[x m :=B] (Tr) 2 and ( ) is obtained by ψ, τ and Lemma 3.3. (Case 2): n 1, and the proof is L hap AB. σ A st (λx 1 x m.m)y 1 y m F 1 F n 1 L st (λx 1 x m.m)y 1 y m F 1 F n. 7 B st F n

8 In this case, we have L hap AB (Case 3): The proof is. i.h. for σ A st M[ y]f 1 F n 1 L st M[ y]f 1 F n. B st F n. σ L hap λz.a A st (λx 1 x m.m)y 1 y m F 1 F n z L st (λx 1 x m.m)y 1 y m F 1 F n where z FV((λx 1 x m.m)y 1 y m F 1 F n ). If M[ y]f 1 F n is not an abstraction, then we have. i.h. for σ L hap λz.a A st M[ y]f 1 F n z L st M[ y]f 1 F n. Note that z FV(M[ y]f 1 F n ) FV((λx 1 x m.m)y 1 y m F 1 F n ). If M[ y]f 1 F n is an abstraction, then n = 0, M = λv.b, and σ is. σ A st (λx 1 x m v.b)y 1 y m z. where v {x 1,..., x m, y 1,..., y m } (otherwise, we rename v). Then we have. i.h. for σ L hap λz.a A st B[ y, z] L st λz.(b[ y, z]) = (λv.b)[ y] where [ y, z] = [x 1 :=y 1,..., x m :=y m, v :=z]. We consider an inference rule: L hap λx.a A st Bx L st B (η + ) with the proviso: x FV(B). This is an extension of rule (B may be an abstraction). Lemma 3.5 The rule (η + ) is admissible; that is, if L hap λx.a, A st Bx, and x FV(B), then L st B. Proof If B is not an abstraction, then (η + ) is just the rule. If B = λy.c, then we have A st (λy.c)x L hap λx.a A st C[y := x] Lem.3.4 L st λx.(c[y := x]) = λy.c. Note that x FV(C). 8

9 Lemma 3.6 If M st N and P st Q, then M[z := P ] st N[z := Q]. Proof Similar to the proof of Lemma 3.3 (y = Q). The only difference is that N[z := Q] may be an abstraction, in Case 5, that is in contravention of the proviso of rule. In such a case, we use the rule (η + ) (Lemma 3.5) to infer M[P ] st N[Q]. Lemma 3.7 If L st (λx.m)nf 1 F n, then L st M[x := N]F 1 F n, where n 0. Proof Similar to the proof of Lemma 3.4. (m = 1 and y m = N. In Case 1, we use Lemma 3.6 instead of 3.3. In Case 3, we use (η + ) rule (Lemma 3.5).) Lemma 3.8 If L st M βη N, then L st N. Proof By induction on the proof of L st M. We divide cases according to the last inference of the proof. Let R be the occurrence of the βη-redex such that M R βη N. (Case 1): The proof is. σ L hap AB A st C L st CD B st D where M = CD. (Subcase 1-1): R = CD; that is, C = λx.e, M = (λx.e)d and N = E[x := D]. This is a special case of Lemma 3.7. (Subcase 1-2): R is in C; that is, C R βη C and N = C D. In this case, we have L hap AB. i.h. for σ A st C L st C D. B st D (Subcase 1-3): R is in D. This is similar to Subcase 1-2. (Case 2): The proof is. σ L hap λx.a A st B L st λx.b where M = λx.b. (Subcase 2-1): R = λx.b; that is, B = Nx, M = λx.(nx), and x FV(N). This is just the rule (η + ) (Lemma 3.5). 9

10 (Subcase 2-2): R is in B; that is, B R βη B, and N = λx.b. Then we have (Case 3): The proof is. i.h. for σ L hap λx.a A st B L st λx.b.. σ L hap λx.a A st Mx L st M where x FV(M) and M is not an abstraction. In this case, we have. i.h. for σ (because Mx βη Nx) L hap λx.a A st Nx (η L st N. + ) (Lem. 3.5) Note that x FV(N) FV(M). Lemma 3.9 M st M. Proof By induction on the structure of M. Theorem 3.10 If M βη N, then M st N. Proof Suppose M = M 1 βη M 2 βη βη M k = N. We can show M st M i for i = 1,..., k, by Lemmas 3.9 and 3.8. Now the Weak Standardization Theorem 2.1 is obvious by Theorems 3.1 and Remarks (1) It seems that we can prove, by the similar proof of Theorem 3.1, the strong version of it: If M st N, then there is a strongly standard βη-reduction sequence from M to N. Then the Strong Standardization Theorem for βη-reduction is proved. (2) In the case of β-reduction, the Leftmost Reduction Theorem is an easy corollary to the Standardization Theorem because we have the following: If M R 1 β Rn β N is a standard β-reduction sequence and N is a β-normal form, then R 1,..., R n are all the leftmost occurrences of β-redexes. However, this does not hold for βηreduction: there are counterexamples ( ) ( ) Ix λx.((λy.(yy))x) λx. (λy.(yy))(ix) βη λx. (λy.(yy))x βη λy.(yy) 10

11 and ( ) I(λy.(yy)) ( ) (λy.(yy))x λx. I(λy.(yy))x βη λx. (λy.(yy))x βη λx.(xx) which are strongly standard and end with βη-normal forms, but neither of them is a leftmost reduction sequence. Note that these can be regarded as the sequences ( ) ( ) Ix (λy.(yy))x λx. (λy.(yy))(ix) βη λx. (λy.(yy))x βη λx.(xx) and ( ) I(λy.(yy)) ( ) λx.((λy.(yy))x) λx. I(λy.(yy))x βη λx. (λy.(yy))x βη λy.(yy), neither of which is weakly ( standard; ) then these are not counterexamples. overlap of redexes λx. (λy.(yy))x causes a problem. The References [1] H.P. Barendregt, The Lambda Calculus (North-Holland, 1984). [2] R. Hindley, Standard and normal reductions, Trans. Amer. Math. Soc. 241 (1978) [3] R. Kashima, A proof of the standardization theorem in λ-calculus, Research Reports on Mathematical and Computing Sciences C-145, (Tokyo Institute of Technology, 2000). [4] J.W. Klop, Combinatory Reduction Systems, Mathematical Center Tracts 127, Amsterdam (1980). [5] G. Mitschke, The standardization theorem for λ-calculus, Zeitshr. Math. Logik Grundlag. Math. 25 (1979) [6] M. Takahashi, Parallel reductions in λ-calculus, Information and Computation 118 (1995) [7] H. Xi, Upper bounds for standardizations and an application, J. Symbolic Logic 64 (1999)